Certainly! Let's simplify the given expression step-by-step.
We start with the logarithmic expression:
[tex]\[
\log_2(x-1) + \log_2(x-3) + \log_4(16)
\][/tex]
### Step 1: Combine the logarithms with base 2
First, we use the logarithm property that states [tex]\(\log_b(a) + \log_b(c) = \log_b(ac)\)[/tex]. Using this property, we can combine the first two logarithms:
[tex]\[
\log_2(x-1) + \log_2(x-3) = \log_2((x-1)(x-3))
\][/tex]
### Step 2: Simplify the logarithm with base 4
Next, consider the term [tex]\(\log_4(16)\)[/tex]. We know that 16 can be written as [tex]\(4^2\)[/tex] since [tex]\(4^2 = 16\)[/tex]. Therefore:
[tex]\[
\log_4(16) = \log_4(4^2)
\][/tex]
Using the property of logarithms that [tex]\(\log_b(b^a) = a \cdot \log_b(b)\)[/tex] and [tex]\(\log_b(b) = 1\)[/tex]:
[tex]\[
\log_4(4^2) = 2 \cdot \log_4(4) = 2 \cdot 1 = 2
\][/tex]
So,
[tex]\[
\log_4(16) = 2
\][/tex]
### Step 3: Combine all the results
Now we need to combine [tex]\(\log_2((x-1)(x-3))\)[/tex] and the constant 2. We can write this in the form:
[tex]\[
\log_2((x-1)(x-3)) + 2
\][/tex]
To combine the logarithmic term and the constant into a single logarithmic expression with base 2, we recognize that adding 2 is equivalent to [tex]\(\log_2(4)\)[/tex] because [tex]\(\log_2(4) = 2\)[/tex].
Thus, we can rewrite the expression as:
[tex]\[
\log_2((x-1)(x-3)) + \log_2(4)
\][/tex]
Using the property [tex]\(\log_b(a) + \log_b(c) = \log_b(ac)\)[/tex] again, we get:
[tex]\[
\log_2((x-1)(x-3) \cdot 4)
\][/tex]
### Final Expression
So, the expression [tex]\(\log_2(x-1) + \log_2(x-3) + \log_4(16)\)[/tex] can be written as a single logarithm:
[tex]\[
\log_2(4(x-1)(x-3))
\][/tex]
Thus, this is the simplified single logarithm form of the given expression.
